In this work, we developed a compartmental bio-mathematical model to
study the effect of treatment in the control of malaria in a population with
infected immigrants. In particular, the vector-host population model consists
of eleven variables, for which graphical profiles were provided to depict their
individual variations with time. This was possible with the help of MathCAD
software which implements the Runge-Kutta numerical algorithm to solve
numerically the eleven differential equations representing the vector-host
malaria population model. We computed the basic reproduction ratio R0 following the next generation matrix. This procedure converts a system of
ordinary differential equations of a model of infectious disease dynamics to
an operator that translates from one generation of infectious individuals to
the next. We obtained R0 = , i.e., the square root of the product of
the basic reproduction ratios for the mosquito and human populations respectively. R0m explains the number of humans that one mosquito can infect
through contact during the life time it survives as infectious. R0h on the
other hand describes the number of mosquitoes that are infected through
contacts with the infectious human during infectious period. Sensitivity
analysis was performed for the parameters of the model to help us know
which parameters in particular have high impact on the disease transmission,
in other words on the basic reproduction ratio R0.
References
[1]
WHO (2010) Estimates of Malaria Cases and Deaths in Africa. 2000-2009.
http://www.rollbackmalaria.org/about-malaria/key-facts
[2]
Matson, A. (1957) The History of Malaria in Nandi. East African Medical Journal, 34, 431-441.
[3]
Johansson, P. and Leander, J. (2010) Mathematical Modeling of Malaria-Methods for Simulation of Epidemics. A Report from Chalmers University of Technology Gothenburg.
[4]
Killeen. G.F. and Smith, T.A. (2007) Exploring the Contributions of Bed Nets, Cattle, Insecticides and Excitorepellency to Malaria Control: A Deterministic Model of Mosquito Host-Seeking Behaviour and Mortality. Transactions of the Royal Society of Tropical Medicine and Hygiene, 101, 867-880.
https://doi.org/10.1016/j.trstmh.2007.04.022
[5]
Yang, H.M. (2000) Malaria Transmission Model for Different Levels of Acquired Immunity and Temperature-Dependent Parameters (Vectors). Revista de Saudepublica, 34, 223-231. https://doi.org/10.1590/S0034-89102000000300003
[6]
Ross, R. (1910) The Prevention of Malaria. J. Murray, London.
[7]
McDonald, G. (1957) The Epidemiology and Control of Malaria. Oxford University Press, London.
[8]
Birhoff, G. and Rotta, G. (1989) Ordinary Differential Equations. 4th Edition.
[9]
Van den Driessche, P. and Watmough, J. (2002) Reproduction Numbers and Sub-Threshold Endemic Equilibria for Compartmental Models of Disease Transmission. Mathematical Biosciences, 180, 29-48.
https://doi.org/10.1016/S0025-5564(02)00108-6
[10]
Hethcote, H.W. (2004) The Mathematics of Infectious Diseases. SIAM Review, 42, 599-653. https://doi.org/10.1137/S0036144500371907
[11]
Iyare, B.S.E., Okuonghae, D. and Osagiede, F.E.U. (2014) A Model for the Transmission Dynamics of Malaria with Infective Immigrants and Its Optimal Control Analysis. Journal of the Association of Mathematical Physics, 28, 163-176.
[12]
Diekmann, O., Heesterbeek, J.A.P. and Metz, J.A.J. (1990) On the Definition and the Computation of the Basic Reproduction Ratio R0 in Models for Infectious Diseases in Heterogeneous Populations. Journal of Mathematical Biology, 28, 365-382.
[13]
Okosun, K.O. and Makinde, O.D. (2011) Modeling the Impact of Drug Resistance in Malaria Transmission and Its Optimal Control Analysis. International Journal of the Physical Science, 28, 6479-6487.
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, i.e., the square root of the product of
the basic reproduction ratios for the mosquito and human populations respectively. R0m explains the number of humans that one mosquito can infect
through contact during the life time it survives as infectious. R0h on the
other hand describes the number of mosquitoes that are infected through
contacts with the infectious human during infectious period. Sensitivity
analysis was performed for the parameters of the model to help us know
which parameters in particular have high impact on the disease transmission,
in other words on the basic reproduction ratio R0.
